In mathematics, the hypograph or subgraph of a function <math>f:\R^{n}\rightarrow \R</math> is the set of points lying on or below its graph.
A related definition is that of such a function's epigraph, which is the set of points on or above the function's graph.
The domain (rather than the codomain) of the function is not particularly important for this definition; it can be an arbitrary set[1] instead of <math>\mathbb{R}^n</math>.
The definition of the hypograph was inspired by that of the graph of a function, where the Шаблон:Em of <math>f : X \to Y</math> is defined to be the set
<math>\operatorname{graph} f := \left\{ (x, y) \in X \times Y ~:~ y = f(x) \right\}.</math>
The Шаблон:Em or Шаблон:Em of a function <math>f : X \to [-\infty, \infty]</math> valued in the extended real numbers <math>[-\infty, \infty] = \mathbb{R} \cup \{ \pm \infty \}</math> is the setШаблон:Sfn
<math>
\begin{alignat}{4}
\operatorname{hyp} f
&= \left\{ (x, r) \in X \times \mathbb{R} ~:~ r \leq f(x) \right\} \\
&= \left[ f^{-1}(\infty) \times \mathbb{R} \right] \cup \bigcup_{x \in f^{-1}(\mathbb{R})} (\{ x \} \times (-\infty, f(x)]).
\end{alignat}
</math>
Similarly, the set of points on or above the function is its epigraph.
Шаблон:Anchor
The Шаблон:Em is the hypograph with the graph removed:
<math>
\begin{alignat}{4}
\operatorname{hyp}_S f
&= \left\{ (x, r) \in X \times \mathbb{R} ~:~ r < f(x) \right\} \\
&= \operatorname{hyp} f \setminus \operatorname{graph} f \\
&= \bigcup_{x \in X} (\{ x \} \times (-\infty, f(x))).
\end{alignat}
</math>
Despite the fact that <math>f</math> might take one (or both) of <math>\pm \infty</math> as a value (in which case its graph would Шаблон:Em be a subset of <math>X \times \mathbb{R}</math>), the hypograph of <math>f</math> is nevertheless defined to be a subset of <math>X \times \mathbb{R}</math> rather than of <math>X \times [-\infty, \infty].</math>
Properties
The hypograph of a function <math>f</math> is empty if and only if <math>f</math> is identically equal to negative infinity.
A function is concave if and only if its hypograph is a convex set. The hypograph of a real affine function <math>g : \mathbb{R}^n \to \mathbb{R}</math> is a halfspace in <math>\mathbb{R}^{n+1}.</math>
